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Overlay Stitch Meshing. A competitive algorithm for no-large-angle triangulation. Don Sheehy Joint work with Gary Miller and Todd Phillips To appear at ICALP 2007. The Problem. Input: A Planar Straight Line Graph. The Problem. Input: A Planar Straight Line Graph.
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Overlay Stitch Meshing Don Sheehy Overlay Stitch Meshing
A competitive algorithm for no-large-angle triangulation Don Sheehy Joint work with Gary Miller and Todd Phillips To appear at ICALP 2007 Don Sheehy Overlay Stitch Meshing
The Problem Input: A Planar Straight Line Graph Don Sheehy Overlay Stitch Meshing
The Problem Input: A Planar Straight Line Graph Output: A Conforming Triangulation Don Sheehy Overlay Stitch Meshing
The Problem Input: A Planar Straight Line Graph Output: A Conforming Triangulation Don Sheehy Overlay Stitch Meshing
Why would you want to do that? Don Sheehy Overlay Stitch Meshing
Don Sheehy Overlay Stitch Meshing
Don Sheehy Overlay Stitch Meshing
Don Sheehy Overlay Stitch Meshing
Don Sheehy Overlay Stitch Meshing
Don Sheehy Overlay Stitch Meshing
What went wrong? Don Sheehy Overlay Stitch Meshing
? ? What if you don’t know the function? Don Sheehy Overlay Stitch Meshing
2 Definitions of Quality 1. No Large Angles [Babuska, Aziz 1976] Don Sheehy Overlay Stitch Meshing
2 Definitions of Quality 1. No Large Angles [Babuska, Aziz 1976] 2. No Small Angles Don Sheehy Overlay Stitch Meshing
No Small Angles Don Sheehy Overlay Stitch Meshing
No Small Angles You may have heard of these before. Delaunay Refinement Sparse Voronoi Refinement Quadtree Don Sheehy Overlay Stitch Meshing
Paying for the spread Spread = L/s L s Don Sheehy Overlay Stitch Meshing
Paying for the spread Optimal No-Large-Angle Triangulation Don Sheehy Overlay Stitch Meshing
Paying for the spread What if we don’t allow small angles? Don Sheehy Overlay Stitch Meshing
Paying for the spread What if we don’t allow small angles? O(L/s) triangles! Don Sheehy Overlay Stitch Meshing
Paying for the spread What if we don’t allow small angles? Fact: For inputs with NO edges, no-small-angle meshing algorithms produce output with O(n log L/s) size and angles between 30o and 120o O(L/s) triangles! Don Sheehy Overlay Stitch Meshing
What to do? Small input angles can force even smaller ouput angles. [Shewchuk ’02] Don Sheehy Overlay Stitch Meshing
No Large Angles Don Sheehy Overlay Stitch Meshing
Polygons with Holes [Bern, Mitchell, Ruppert 95] – - All triangles are nonobtuse. - Output has O(n) triangles. Don Sheehy Overlay Stitch Meshing
Polygons with Holes [Bern, Mitchell, Ruppert 95] – - All triangles are nonobtuse. - Output has O(n) triangles. Does not work for arbitrary PSLGs Don Sheehy Overlay Stitch Meshing
Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing
Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing
Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing
Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing
Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing
Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing
Propagating Paths Don Sheehy Overlay Stitch Meshing
Propagating Paths Don Sheehy Overlay Stitch Meshing
Propagating Paths First introduced by Mitchell [93] Later Improved by Tan [96] Worst Case Optimal Size O(n2) Angle bounds: 132o Don Sheehy Overlay Stitch Meshing
The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing
The OSM Algorithm(Overlay Stitch Meshing) • Results: • Angles bounded by 170o • O(log L/s) competitive size Don Sheehy Overlay Stitch Meshing
The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing
The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing
The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing
The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing
The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing
The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing
The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing
The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing
The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing
The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing
The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing
The OSM Algorithm(Overlay Stitch Meshing) • An Overlay Edge is kept if 1. It does not intersect the input, OR 2. It forms a good intersection with the input. at least 30o Don Sheehy Overlay Stitch Meshing
Angle Guarantees Don Sheehy Overlay Stitch Meshing